Find the order of elements 5, 7 and 17 in $\{ G,{\text{ }}{ \times _{18}}\}$.

State whether or not $\{ G,{\text{ }}{ \times _{18}}\}$ is cyclic, justifying your answer.

Write down the elements in set $K$.

Find the left cosets of $K$ in $\{ G,{\text{ }}{ \times _{18}}\}$.

considering powers of elements     (M1)

5 has order 6     A1

7 has order 3     A1

17 has order 2     A1

[4 marks]

$G$ is cyclic     A1

because there is an element (are elements) of order 6     R1

Note:     Accept “there is a generator”; allow A1R0.

[3 marks]

$\{ 1,{\text{ }}17\}$     A1

[1 mark]

multiplying $\{ 1,{\text{ }}17\}$ by each element of $G$     (M1)

$\{ 1,{\text{ }}17\} ,{\text{ }}\{ 5,{\text{ }}13\} ,{\text{ }}\{ 7,{\text{ }}11\}$     A1A1A1

[4 marks]